The Gamma Factor

Imagine a clock that worked by reflecting a pulse of light off two mirrors a distance L apart.

There are two of these clocks. One stationary and one on a fast moving lorry with velocity v. On the animation you should see that the light pulse on the moving clock travels further in the same time than the pulse on the stationary clock.

Of course this does not take into account the constancy of the speed of light. We now know that the speed of the light pulses is the same. It is time for the moving object which runs slower.

How much slower is called the gamma factor, γ, and that is what we are going to calculate now. Remember that γ is a number greater than 1.

The triangle shows one tick of the clock on the lorry as seen by a stationary observer.

In time T the light pulse travels a distance cT, the hypotenuese

The lorry travels a distance vT

For someone on the lorry the pulse travels distance L = cT'

As time runs slower on the lorry T is bigger than T' (i.e. more time will pass for the stationary observer) and the ratio of the two is γ

Using this expression we can calculate how much slower time runs for the moving observer.

Example 1

A proton in a particle accelerator travels at 0.99 times the speed of light. To us the proton appears to take 9.1 x 10^-5 seconds to travel the 27km circumference of the accelerator. If you were the proton how long would it seem to take?

Example 2

The travelling twin (in the twin paradox) ages 5 years on a journey travelling at 0.9c. How old is the twin that remained on Earth?